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Orderings

Sometimes we will write, for a relation $R$, $xRy$ instead of $(x,y)\in R$. In this chapter we will deal with ordering -- relations with special properties and we will denote these relations usally $\leq$. In fact, the definition of ordering reminds properties of the usual relation $\leq$ on numbers.

A relation R on set A is called

• reflexive iff aRa for any $a\in A$;
• antisymmetric iff if aRb and bRa implies a = b for any $a,b\in A$;
• transitive iff aRb and bRc implies aRc for any $a,b\in A$.

partial order

Note about weak < and strict partial order.

totally ordered set linearly ordered set (total order, linear order), chain

antichain

Examples: $\subseteq$, |

minimal element

smallest element

well-ordering

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See also

• Partially ordered set at Wikipedia

This is incomplete and a draft, like most wikibooks, additional information is to be added